kung,rota - combinatorics - the rota way (cambridge, 2009)

This page intentionally left blank P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Combinatorics: The Rota Way Gian-Carlo Rota was one of the most original and colorful mathematicians of the twentieth century. His work on the foundations of combinatorics focused on revealing the algebraic structures that lie behind diverse combinatorial areas and created a new area of algebraic combinatorics. His graduate courses influenced generations of students. Written by two of his former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially or- dered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, M ¨ obius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exer- cises and research problems are included and unexplored areas of possible research are discussed. This book should be on the shelf of all students and researchers in combinatorics and related areas. joseph p. s. kung is a professor of mathematics at the University of North Texas. He is currently an editor-in-chief of Advances in Applied Mathematics. gian-carlo rota (1932–1999) was a professor of applied mathematics and natural philosophy at the Massachusetts Institute of Technology. He was a member of the Na- tional Academy of Science. Hewas awarded the 1988 Steele Prize of the American Math- ematical Society for his 1964 paper “On the Foundations of Combinatorial Theory I. Theory of M ¨ obius Functions.” He was a founding editor of Journal of Combinatorial Theory. catherine h. yan is a professor of mathematics at Texas A&M University. Prior to that, she was a Courant Instructor at New York University and a Sloan Fellow. i P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 ii P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Cambridge Mathematical Library Cambridge University Press has a long and honorable history of publishing in mathematics and counts many classics of the mathematical literature within its list. Some of these titles have been out of print for many years now and yet the methods they espouse are still of considerable relevance today. The C ambridge Mathematical Library will provide an inexpensive edition of these titles in a durable paperback format and at a price that will make the books attractive to individuals wishing to add them to their personal libraries. It is intended that certain volumes in the series will have forewords, written by leading experts in the s ubject, which will place the title in its historical and mathematical context. iii P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Gian-Carlo Rota, Circa 1970 Pencil drawing by Eleanor Blair iv P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Combinatorics: The Rota Way JOSEPH P. S. KUNG University of North Texas GIAN-CARLO ROTA CATHERINE H. YAN Texas A&M University v CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-88389-4 ISBN-13 978-0-521-73794-4 ISBN-13 978-0-511-50687-1 © Joseph P. S. Kung, Gian-Carlo Rota, and Catherine H. Yan 2009 2009 Information on this title: www.cambrid g e.or g /9780521883894 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org p a p erback eBook ( EBL ) hardback P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Contents Preface page ix 1 Sets, Functions, and Relations 1 1.1. Sets, Valuations, and Boolean Algebras 1 1.2. Partially Ordered Sets 9 1.3. Lattices 17 1.4. Functions, Partitions, and Entropy 28 1.5. Relations 44 1.6. Further Reading 52 2 Matching Theory 53 2.1. What Is Matching Theory? 53 2.2. The Marriage Theorem 54 2.3. Free and Incidence Matrices 62 2.4. Submodular Functions and Independent Matchings 67 2.5. Rado’s Theorem on Subrelations 74 2.6. Doubly Stochastic Matrices 78 2.7. The Gale-Ryser Theorem 94 2.8. Matching Theory in Higher Dimensions 101 2.9. Further Reading 105 3 Partially Ordered Sets and Lattices 106 3.1. M ¨ obius Functions 106 3.2. Chains and Antichains 126 3.3. Sperner Theory 136 3.4. Modular and Linear Lattices 147 3.5. Finite Modular and Geometric Lattices 161 3.6. Valuation Rings and M ¨ obius Algebras 171 3.7. Further Reading 176 vii P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 viii Contents 4 Generating Functions and the Umbral Calculus 178 4.1. Generating Functions 178 4.2. Elementary Umbral Calculus 185 4.3. Polynomial Sequences of Binomial Type 188 4.4. Sheffer Sequences 205 4.5. Umbral Composition and Connection Matrices 211 4.6. The Riemann Zeta Function 218 5 Symmetric Functions and Baxter Algebras 222 5.1. Symmetric Functions 222 5.2. Distribution, Occupancy, and the Partition Lattice 225 5.3. Enumeration Under a Group Action 235 5.4. Baxter Operators 242 5.5. Free Baxter Algebras 246 5.6. Identities in Baxter Algebras 253 5.7. Symmetric Functions Over Finite Fields 259 5.8. Historical Remarks and Further Reading 270 6 Determinants, Matrices, and Polynomials 272 6.1. Polynomials 272 6.2. Apolarity 278 6.3. Grace’s Theorem 283 6.4. Multiplier Sequences 291 6.5. Totally Positive Matrices 296 6.6. Exterior Algebras and Compound Matrices 303 6.7. Eigenvalues of Totally Positive Matrices 311 6.8. Variation Decreasing Matrices 314 6.9. P ´ olya Frequency Sequences 317 Selected Solutions 324 Bibliography 369 Index 389 [...]... refer to the third author simply as Rota As for the title, we wanted one that is not boring The word way is not meant to be prescriptive, in the sense of “my way or the highway.” Rather, it comes from the core of the cultures of the three authors The word way resonates with the word “cammin” in the first line of Dante’s Divina commedia, “Nel mezzo del cammin di nostra vita.” It also resonates as the. .. that each of the four properties is equivalent to being a well-quasiorder (d) Let Q be a well-quasi-order Show that if P ⊆ Q, then P (with quasiorder inherited from Q) is a well-quasi-order Show that if P is a quasi-order and there exists a quasi-order-preserving map Q → P , then P is a well-quasiorder (e) An induction principle For an element a in a quasi-order set Q, let F (a) = {x: x ≥ a}, the principal... generated by a Show that if the complements Q\F (a) are well-quasi-orders for all a ∈ Q, then Q itself is a wellquasi-order (f) Show that if P and Q are well-quasi-orders, then the Cartesian product P × Q is a well-quasi-order The set N of nonnegative integers, ordered so that 0 < 1 < 2 < 3 < · · · , is a well-quasi-order From (f), one derives immediately Gordan’s lemma: the Cartesian product N × N... CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Preface The working title of this book was Combinatorics 18.315.” In the private language of the Massachusetts Institute of Technology, Course 18 is Mathematics, and 18.315 is the beginning graduate course in combinatorial theory From the 1960s to the 1990s, 18.315 was taught primarily by the three permanent faculty in combinatorics, Gian-Carlo... senses, the way has to be struggled for and sought individually This is best expressed in Chinese: Inadequately translated into rectilinear English, this says “a way which can be wayed (that is, taught or followed) cannot be a way. ” Rota s way is but one way of doing combinatorics After “seeing through” Rota s way, the reader will seek his or her own way It is our duty and pleasure to thank the many... xi in Q, then x1 x2 x3 x4 ≤ y1 y2 x4 x1 y3 y2 y2 x2 x3 x1 y4 y5 x4 x1 y6 y7 The quasi-order Set(Q) is the set of all finite subsets of Q, quasi-ordered by A ≤ B if there is an injection f : A → B such that a ≤ f (a) for every a ∈ A (h) Prove Higman’s lemma If Q is a well-quasi-order, then Seq(Q) and Set(Q) are well-quasi-orders Higman’s lemma gives the most useful cases of a more general theorem, also... does not contain any of the graphs M1 , M2 , , Mr as minors This very general theorem is in fact true, and follows from the fact that the set of finite graphs, ordered by being a minor, is a well-quasi-order (j) The Robertson–Seymour graph minor theorem Show that the set of all finite graphs ordered under minors is a well-quasi-order (k) The matroid minor project Show that the set L(q) of all matroids... order-preserving functions from P to the Boolean algebra 2: the ideal I corresponds to the function f : P → 2 defined ˆ ˆ by f (x) = 0 if x ∈ I and f (x) = 1 otherwise Filters are “up-closed;” in other words, filters are ideals in the order-dual P ↓ The set complement P \I of an ideal is a filter Any statement about ideals inverts to a statement about filters In particular, the map sending a filter to the. .. finite, a quasi-order defines a topology and, conversely, a topology defines a quasi-order (b) Show that the relation x ∼ y if x ≤ y and y ≤ x is an equivalence relation on Q Define a natural partial order on the equivalence classes from ≤ A quasi-order Q is a well-quasi-order (often abbreviated to wqo) if there are no infinite antichains or infinite descending chains The property of being well-quasi-ordered... selected papers; these P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 Preface November 28, 2008 12:21 xi are referenced by short titles Our convention is explained in the beginning of the bibliography We should now explain the authorship and the title of this book Gian-Carlo Rota passed away unexpectedly in 1999, a week before his 67th birthday This book was physically written by the two authors signing . Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 97 8-0 -5 2 1-8 838 9-4 ISBN-13 97 8-0 -5 2 1-7 379 4-4 ISBN-13 97 8-0 -5 1 1-5 068 7-1 © Joseph P. S. Kung, Gian-Carlo Rota, . prescriptive, in the sense of “my way or the highway.” Rather, it comes from the core of the cultures of the three authors. The word way resonates with the word “cammin” in the first line of. blank P1: CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Combinatorics: The Rota Way Gian-Carlo Rota was one of the most original and colorful mathematicians of the twentieth